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Graph Connectivities, Network Coding, and Expander Graphs∗ SIAM J. COMPUT. c 2013 Society for Industrial and Applied Mathematics Vol. 42, No. 3, pp. 733–751 GRAPH CONNECTIVITIES, NETWORK CODING, AND EXPANDER GRAPHS∗ HO YEE CHEUNG†,LAPCHILAU†, AND KAI MAN LEUNG† Abstract. We present a new algebraic formulation for computing edge connectivities in a directed graph, using ideas developed in network coding. This reduces the problem of computing edge connectivities to solving systems of linear equations, thus allowing us to use tools in linear algebra to design new algorithms. Using the algebraic formulation, we obtain faster algorithms for computing single source edge connectivities and all pairs edge connectivities. In some settings, the amortized time to compute the edge connectivity for one pair is sublinear. Through this connection, we have also found an interesting use of expanders and superconcentrators to design fast algorithms for some graph connectivity problems. Key words. graph connectivities, network coding, expander graphs, linear equations, random- ized algorithms AMS subject classifications. 05C40, 05C50, 05C85, 68Q25, 68W20 DOI. 10.1137/110844970 1. Introduction. Graph connectivity is a basic concept that measures the reli- ability and efficiency of a graph. The edge connectivity from vertex s to vertex t is defined as the size of a minimum s-t cut or, equivalently, the maximum number of edge disjoint paths from s to t. Computing edge connectivities is a classical and well- studied problem in combinatorial optimization. Most known algorithms for solving this problem are based on network flow techniques (see, e.g., [34]). Even and Tarjan [12] introduced the fastest algorithm to compute s-t edge con- nectivity in a simple directed graph and running in O(min{m1/2,n2/3}·m)time, where m is the number of edges and n is the number of vertices. To compute the edge connectivities for many pairs, however, it is not known how to do it faster than computing edge connectivity for each pair separately, even when the pairs share the source or the sink. For instance, it is not known how to compute all pairs edge con- nectivities faster than computing s-t edge connectivity for Θ(n2)pairs.Thisisin contrast to the problem in undirected graphs, where all pairs edge connectivities can be computed in O˜(mn) time by constructing a Gomory–Hu tree [6]. Network coding is an innovative method for transmitting information in a com- puter network. The fundamental result is a max-information-flow min-cut theorem for multicasting [1]: if the edge connectivity from the source vertex s to each sink vertex ti is at least k, then one can transmit k units of information to all sink vertices simultaneously by performing encoding and decoding at the vertices. An elegant alge- braic framework has been developed for constructing efficient network coding schemes for multicasting [29, 26]. In this paper, we use the techniques developed in network coding to obtain a new algebraic formulation for computing edge connectivities. This reduces the problem ∗Received by the editors August 19, 2011; accepted for publication (in revised form) December 26, 2012; published electronically May 2, 2013. A preliminary version of this paper appeared in Proceedings of the 52nd Annual IEEE Symposium on Foundations of Computer Science, 2011. http://www.siam.org/journals/sicomp/42-3/84497.html †The Chinese University of Hong Kong, Shatin, N.7., Hong Kong ([email protected], [email protected], [email protected]). The second author’s research was supported by HK RGC grants 413609 and 413411. 733 734 HO YEE CHEUNG, LAP CHI LAU, AND KAI MAN LEUNG of computing edge connectivities to solving systems of linear equations and opens up new directions in the design of algorithms for the problem. One advantage of the technique is that the edge connectivities from a source vertex to all other vertices can be computed simultaneously (as in the max-information-flow min-cut theorem). This leads to faster algorithms for computing single source edge connectivities and all pairs edge connectivities, and in some settings the amortized time to compute the edge connectivity for one pair is sublinear. In the process, we have also found an interesting use of expanders and superconcentrators to design fast algorithms for graph connectivity problems. 1.1. Our results. Our new algebraic formulation for computing edge connectiv- ities is inspired by the random linear coding algorithm [18] for constructing network codes. Let G =(V,E) be a directed graph. Let s be the source vertex with out degree d, with outgoing edges e1,...,ed.Toeachedgee ∈ E we associate a vector fe of dimension d, where each entry in fe is an element from a large enough finite field F. The vectors fe are required to satisfy the following properties: (1) the vectors on e1,...,ed are linearly independent, and (2) the vector on an edge e =(v, w)isa random linear combination of the incoming vectors of v. Once we obtain the vectors, we can compute the edge connectivities from the source vertex as follows. Theorem 1.1 (informal statement). With high probability there is a unique solution to the vectors, and the edge connectivity from s to t is equal to the rank of theincomingvectorsoft for any t ∈ V − s. See Figure 1.1 for an example and Theorem 2.1 for the formal statement. This formulation was previously known for only directed acyclic graphs [22, 36]. For general Fig. 1.1. In this example, three independent vectors f1, f2 and f3 are sent from the source s. Other vectors are a linear combination of the incoming vectors, according to the random coefficients on the dotted lines. All operations are done in the field of size 11. To compute the edge connectivity from s to v2, for instance, we compute the rank of (f2,f4,f5) which is 3 in this example. GRAPH CONNECTIVITIES, NETWORK CODING, EXPANDERS 735 directed graphs, network coding schemes require convolution codes [11, 28], and it is not known how to use these to compute edge connectivities. Our contribution is a simple formulation that can be used to compute edge connectivities. The algebraic formulation reduces the problem of computing edge connectivities to solving systems of linear equations. We call the step to compute the vectors fe the encoding step and call the step to compute the ranks the decoding step. The formulation has the advantage that after the encoding step has been done, the edge connectivities from the source vertex s to all other vertices can be computed readily. We first present algorithmic results on single source edge connectivities and then algorithmic results on all pairs edge connectivities. 1.1.1. Single source edge connectivities. In directed acyclic graphs, the en- coding step can be implemented directly without solving linear equations, but the resulting algorithm is not competitive with the existing algorithms. By a simple transformation using superconcentrators [37, 19], we show how to implement the en- coding step optimally by this direct approach. This can be used to improve the random linear coding algorithm [18] for network coding. Also we obtain a faster algo- rithm for computing single source edge connectivities in directed acyclic graphs. The algorithm runs in linear time when d is a constant, and by a simple reduction this can be used to return all vertices with edge connectivity at most d from the source s in linear time. Theorem 1.2. Given a simple directed acyclic graph and a source vertex s with out-degree d, the encoding step can be done in O(dm) time, and the edge connectivities from s to all vertices can be computed in O(dω−1m) time with high probability, where ω ≈ 2.38 is the matrix multiplication exponent. In some graphs, the system of linear equations can be solved more efficiently. For instance, we can use a recent result of Alon and Yuster [3] to perform the encod- ing step faster in directed planar graphs with constant maximum degree. The best known algorithm for computing single source edge connectivities in directed planar graphs requires O(n2)timebyusinganO(n) time algorithm to compute s-t edge connectivity [8]. Theorem 1.3. Given a simple directed planar graph with constant maximum de- gree and a source vertex s, the edge connectivity from s to all vertices can be computed in O(nω/2) time with high probability. 1.1.2. All pairs edge connectivities. For general directed graphs, we show that all pairs edge connectivities can be computed in one matrix inversion time, in- stead of solving the linear equations for each source vertex separately. The algorithm is faster when the graph has O(n1.93) edges. For example, when m = O(n), it takes O(nω) time, while the best known algorithm takes O(n3.5)time. Theorem 1.4. Given a simple directed graph, the edge connectivities between all pairs of vertices can be computed in O(mω) time with high probability, where m is the number of edges in the graph. We show that the matrix inversion can be computed more efficiently for graphs that are “well-separable” (which will be defined in section 4.2), which includes planar graphs, bounded genus graphs, fixed minor free graphs, etc. The resulting√ algorithm is faster than that for general graphs when the maximum degree is O( n). Theorem 1.5. Given a simple well-separable directed graph with maximum degree d, the edge connectivities between all pairs of vertices can be computed in O(dω−2nω/2+1) time with high probability. 736 HO YEE CHEUNG, LAP CHI LAU, AND KAI MAN LEUNG 1.1.3. Edge splitting-off. The idea of transforming the graph using supercon- centrators can also be used in other graph connectivity problems.
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